I've played multiple worlds so far with this mod, and only the common and acid biters spawn. I have all the variation boxes checked, and it still only spawns common biters. Any way to fix?

As I see new factions appear via two mechanisms (and it works!):

1. expanding occupied area. Then some parts of players base (some outposts for example) will be located in areas of different enemy clans - different factions usually, but if you explored a huge map very early you can be unlucky and get only Common and Acid at very low evolution level (if you discovered all land at evolution 5% for example). Don't worry - this mechanism is too slow anyway to see any change.

2. Mutations. Every clan can mutate up to a number of times specified in mod map settings (default value is 2, I recommend to increase it to some much greater value). Mutation is a change of main faction (you called it "variation"). Mutations occur
   a) once player kills number of biters - see "AI: Adaptation Modifier" mod map setting. It selects a faction with some resistant to damage type used to kill most of them (usually first type is physical, then it will be fire/laser, etc).
   b) random mutations. I don't know what triggers them, but they do consume "Max Mutations per faction" too (wrong wording in setting name - it's not faction, but some other name like "baseAI" or tribe/clan/kindred).
   Mutation mechanism is very fast. After mutation occurred all new/upgraded nests will be of that new faction. Old nests/worms change only via slow upgrade. Biters faction is determined by nest faction, but when forming a squad or even when walking around nests they effectively mix.

So how much time did you play? Mutation should occur sooner or later. It happens many times in my playthroughs.

I played for about 30-50 hours on a few worlds, only one of them I saw electric biters spawn near the end. Maybe I just need to play longer or turn up the adaptation modifier?

I also tried loading up a sandbox world, speeding up the game 10x, setting the evolution to 100%, and killed a bunch of them with a spidertron and I didn't see any new mutations spawning. Is there any way I can test the mutations?

or turn up the adaptation modifier?

turn down this modifier or kill more biters (4500 biters required by default. I have to kill 25k by 20% evolution on medium map start difficulty and 40k by the same moment on high difficulty, therefore I have to triple this modifier to avoid mutations taking place too early)

I also tried loading up a sandbox world, speeding up the game 10x, setting the evolution to 100%, and killed a bunch of them with a spidertron and I didn't see any new mutations spawning. Is there any way I can test the mutations?

Higher evolution demands even more biters to be killed for adaptation mutation to happen. Set evolution on some early-mid game value like 30% (not less than 20 and not higher than 45%), set AI:Adaption Modifier to 0.001 , save map settings, check that mod settings for current map were saved correctly and go kill 10 biters.

Just tested in Sandbox: set evo 45%, very low adaption modifier, killed 350 biters with fire ammo ("Incendiary Magazine") and in 5..7 minutes mutation occurred, caused by fire damage and leading to appearance of... Fast biters :-) I don't know why. Probably at 100% evo killing 1000 biters will give more logical resistant faction generation. I assume (and I hope) that some factions (like Poison, Nuclear, Laser) appear only after some evo value reached no matter what damage type was used to annihilate enemies.